Assuming the central node is at level 0 (root), this graph becomes a tree. Now it is easy reason about the different types of trees that can be formed and different types of social networks that can be modelled.
In order for the graph to be a tree, communication between nodes has to be restricted, such that the tree structure is not violated (e.g. two nodes on the same level cannot communicate). I can't think of any social network where communication is naturally restricted in this way, but such a social network can be created artificially.
Examples of such artificial social networks would commonly follow a chain-of-command structure. For example in a military setting, the official information flows would have a tree structure.